Ch 21 Magnetic Fields and Forces

Chapter 21Magnetic Forces and Magnetic Fields

AP Learning ObjectivesMagnetic Fields Forces on moving charges in magnetic fields

Students should understand the force experienced by a charged particle in a magnetic field, so they can:

Calculate the magnitude and direction of the force in terms of q, v, and, B, and explain why the magnetic force can perform no work.

Deduce the direction of a magnetic field from information about the forces experienced by charged particles moving through that field.

Describe the paths of charged particles moving in uniform magnetic fields.

Derive and apply the formula for the radius of the circular path of a charge that moves perpendicular to a uniform magnetic field.

Describe under what conditions particles will move with constant velocity through crossed electric and magnetic fields.

AP Learning Objectives Forces on current-carrying wires in magnetic fields

Students should understand the force exerted on a current-carrying wire in a magnetic field, so they can: Calculate the magnitude and direction of the force on a straight

segment of current-carrying wire in a uniform magnetic field. Indicate the direction of magnetic forces on a current-carrying

loop of wire in a magnetic field, and determine how the loop will tend to rotate as a consequence of these forces.

Fields of long current-carrying wiresStudents should understand the magnetic field produced by a long straight current-carrying wire, so they can: Calculate the magnitude and direction of the field at a point in the

vicinity of such a wire. Use superposition to determine the magnetic field produced by

two long wires. Calculate the force of attraction or repulsion between two long

current-carrying wires.

Table Of Contents1. Magnetic Fields

2. The Force That a Magnetic Field Exerts on a Moving Charge

3. The Motion of a Charged Particle in a Magnetic Field

4. The Mass Spectrometer (Not AP-B)

5. The Force on a Current in a Magnetic Field

6. The Torque on a Current-Carrying Coil (Not AP-B)

7. Magnetic Fields Produced by Currents (Straight Wires Only)

8. Ampère’s Law (Not AP-B)

9. Magnetic Materials (Not AP-B)

Chapter 21: Magnetic Forces and

Magnetic FieldsSection 1:

Magnetic Fields

Magnets and Magnetic Fields Magnets cause space to be modified in their vicinity, forming

a “magnetic field”.

The magnetic field caused by magnetic “poles” is analogous

to the electric field caused by electric “poles” or “charges”.

The north pole is where the magnetic field lines leave the

magnet, and the south pole is where they reenter.

Magnetic field lines differ from electric field lines in that

they are continuous loops with no beginning or end.

Magnetic “Monopoles”

Do not exist!

This is another way that magnetic fields differ from

electric fields.

Magnetic poles cannot be separated from each other in the

same way that electric poles (charges) can be.

The needle of a compass is permanent magnet that has a north magnetic pole (N) at one end and a south magnetic pole (S) atthe other.

The behavior of magneticpoles is similar to that oflike and unlike electric charges.

Surrounding a magnet there is a magnetic field. The directionof the magnetic field at any point in space is the direction indicatedby the north pole of a small compass needle placed at that point.

Magnetic Fields

Units of Magnetic Field

Tesla (SI)

N/(C m/s)

N/(A m)

Gauss

1 Tesla = 104 gauss

The magnetic field lines and pattern of iron filings in the vicinity of abar magnet and the magnetic field lines in the gap of a horseshoe magnet.

Magnetic Field Lines

Earth’s Magnetic Field Lines

21.1.1. Consider the two rectangular areas shown with a point P located at the midpoint between the two areas. The rectangular area on the left contains a bar magnet with the south pole near point P. The rectangle on the right is initially empty. How will the magnetic field at P change, if at all, when a second bar magnet is placed on the right rectangle with its south pole near point P?

a) The direction of the magnetic field will not change, but its magnitude will decrease.

b) The direction of the magnetic field will not change, but its magnitude will increase.

c) The magnetic field at P will be zero tesla.

d) The direction of the magnetic field will change and its magnitude will increase.

e) The direction of the magnetic field will change and its magnitude will decrease.

21.1.2. Consider the two rectangular areas shown with a point P located at the midpoint between the two areas. The rectangular area on the left contains a bar magnet with the south pole near point P. The rectangle on the right is initially empty. How will the magnetic field at P change, if at all, when a second bar magnet is placed on the right rectangle with its north pole near point P?

a) The direction of the magnetic field will not change, but its magnitude will decrease.

b) The direction of the magnetic field will not change, but its magnitude will increase.

c) The magnetic field at P will be zero tesla.

d) The direction of the magnetic field will change and its magnitude will increase.

e) The direction of the magnetic field will change and its magnitude will decrease.

21.1.3. What is the direction of the magnetic field at the point P, directly below a point at the center of the magnet? The numbered arrows represent various directions. Direction “1” is to the right, “2” to the left, “3” is upward, “4” is downward, and “5” is toward you.

a) 1

b) 2

c) 3

d) 4

e) 5

21.1.4. Two rods are resting on a table. Although they appear to be identical, one is a permanent magnet and the other is made from soft iron and is not permanently magnetized. Which one of the following methods is most likely to reveal which rod is the magnet and which is the soft iron?

a) Take one of the rods and touch it to each end of the other rod.

b) Use a magnetic monopole to find the end of one of the rods that repels it.

c) Move a compass along each rod to see if the compass needle behaves as it should in a magnetic field.

d) There is no way to tell the difference between the two rods.

21.1.5. You are given a bar magnet, but the poles are not labeled. Using which of the following items can you determine which end is a north pole and which is a south pole?

a) a voltmeter

b) a bottle of iron particles

c) a charged rubber rod

d) a compass

e) a steel rod



The Force That a Magnetic Field Exerts on a Moving Charge

Magnetic Force on Particles Magnetic fields cause the existence of magnetic forces. A magnetic force is exerted on a particle within a magnetic

field only if the particle has a charge. the charged particle is moving with at least a portion of its

velocity perpendicular to the magnetic field.

When a charge is placed in an electric field, it experiences a force, according to

EF

q

The Force That a Magnetic Field Exerts on a Charge

Magnetic Force on a Charged Particle magnitude:

q: charge in Coulombs

v: speed in meters/second

B: magnetic field in Tesla

: angle between v and B

direction: Right Hand Rule

FB = q v x B (This is a “vector cross product”)

SI Unit of Magnetic Field:

sinBvqFB

T tesla1metercoulomb

secondnewton

Common Right Hand Rule

The Right Hand rule to Determine a Vector Cross Product

1. Align your hand (pointer finger) along the first vector.

2. Orient your wrist so that you can “cross” your hand into the

second vector (middle finger).

3. Your thumb gives you the direction of the third vector (which

is the result).

Book’s Right Hand Rule No. 1. Extend the right hand so the fingers point along the direction

of the magnetic field and the thumb points along the velocity of the charge. The palm of the hand then faces in the direction of the magnetic force that acts on a positive charge.

If the moving charge is

negative, the direction of the

force is opposite to that

predicted by RHR-1.

Example 1 Magnetic Forces on Charged ParticlesA proton in a particle accelerator has a speed of 5.0x106 m/s. The proton encounters a magnetic field whose magnitude is 0.40 T and whose directionmakes and angle of 30.0 degrees with respect to the proton’s velocity(see part (c) of the figure). Find (a) the magnitude and direction of the force on the proton and (b) the acceleration of the proton. (c) What wouldbe the force and acceleration of the particle were an electron?

sinvBqF o(a)

(b)

(c)

pm

Fa

Magnitude of the force is the same, but direction is opposite.

em

Fa

N106.1 13F

0.30sinT40.0sm100.5C1060.1 619

213 sm106.9 kg1067.1

N106.127

13

217 sm108.1 kg1011.9

N106.131

13

21.2.1. Which one of the following statements concerning the magnetic force on a charged particle in a magnetic field is true?

a) The magnitude of the force is largest when the particle is not moving.

b) The force is zero if the particle moves perpendicular to the field.

c) The magnitude of the force is largest when the particle moves parallel to the direction of the magnetic field.

d) The force depends on the component of the particle's velocity that is perpendicular to the field.

e) The force acts in the direction of motion for a positively charged particle.

21.2.2. An electron traveling due east in a region that contains only a magnetic field experiences a vertically downward force, toward the surface of the earth. What is the direction of the magnetic field?

a) upward, away from the earth

b) downward, toward the earth

c) due north

d) due west

e) due south

21.2.3. A charged particle is moving through a constant magnetic field. Does the magnetic field do work on the charged particle?

a) yes, because the force is acting as the particle is moving through some distance

b) no, because the magnetic force is always perpendicular to the velocity of the particle

c) no, because the magnetic field is a vector and work is a scalar quantity

d) no, because the magnetic field is conservative

e) no, because the magnetic force is a velocity-dependent force



The Motion of a Charged Particle in a Magnetic Field

Magnetic forces…

are always orthogonal (at right angles) to the plane

established by the velocity and magnetic field vectors.

can accelerate charged particles by changing their direction.

can cause charged particles to move in circular or helical

paths.

Magnetic forces cannot...

change the speed or kinetic energy of charged particles.

do work on charged particles.

Magnetic Forces…

…are centripetal.

Remember that centripetal acceleration is v2/r.

Remember centripetal force is therefore mv2/r.

Charged particle in an electric field.

Charged particle in a magnetic field.

Particle Motion in E & M Fields

Conceptual Example 2 A Velocity Selector

A velocity selector is a device for measuring the velocity of a charged particle. The device operates by applying electric and magnetic forces to the particle in such a way that these forces balance.

How should an electric field be applied so thatthe force it applies to the particle can balancethe magnetic force?

Using the right hand rule and looking only atthe force created by the magnetic field, the particle would be directed: up

Therefore the electric field must be directed: down

The electrical force can do work on acharged particle.

The magnetic force cannot do work on acharged particle.

Work on particles

The magnetic force always remainsperpendicular to the velocity and is directed toward the center of the circular path.

r

vmFc

2

r

vmqvB

2

qB

mvr

Radius of circular motion

Conceptual Example 4 Particle Tracks in a Bubble Chamber

The figure shows the bubble-chamber tracks from an event that begins at pointA. At this point a gamma ray travels in from the left, spontaneously transformsinto two charged particles. The particlesmove away from point A, producing two spiral tracks. A third charged particle isknocked out of a hydrogen atom and moves forward, producing the long track.

The magnetic field is directed out of thepaper. Determine the sign of each particleand which particle is moving most rapidly.

Particle 1:

Particle 2:

Particle 3:

Fastest:

21.3.1. An alpha particle (a helium nucleus which has a net positive charge) is moving due east when it enters a magnetic field that is directed due north. Which one of the following statements best describes the motion of the alpha particle after entering the magnetic field?

a) The particle decelerates while traveling along a straight line until it stops.

b) The particle continues at a constant speed, but its direction changes as it follows a circular path.

c) The particle continues at a constant speed, but its direction changes as it follows a parabolic path.

d) The particle slows and changes direction to accelerate to move due north.

e) The particle slows and changes direction to accelerate to move directly upward.

21.3.2. An electron is traveling due south in a region of space at a constant speed. What can you conclude from this situation regarding the presence any electric and/or magnetic fields?

a) The electric field must be zero, but the magnetic field might be non-zero in the region.

b) The magnetic field must be zero, but the electric field might be non-zero in the region.

c) Both the electric and magnetic field might be non-zero, but they are perpendicular to each other in the region.

d) Both the electric and magnetic field might be non-zero, but they point in opposite directions in the region.

e) Both the electric and magnetic field must be zero in the region.

21.3.3. A positively-charged particle is stationary in a constant magnetic field within a region of space. Which one of the following statements concerning the particle is true?

a) The particle will not move.

b) The particle will accelerate in the direction perpendicular to the field.

c) The particle will accelerate in the direction parallel to the field.

d) The particle will accelerate in the direction opposite to the field.

e) The particle will move with constant velocity in the direction of the field.

21.3.4. A negatively-charged particle travels parallel to magnetic field lines within a region of space. Which one of the following statements concerning the force exerted on the particle is true?

a) The force is directed perpendicular to the magnetic field.

b) The force is perpendicular to the direction in which the particle is moving.

c) The force slows the particle.

d) The force accelerates the particle.

e) The force has a magnitude of zero newtons.



The Mass Spectrometer (Not AP-B)

eB

mv

qB

mvr eVmv 2

21

magnitude ofelectron charge

KE=PE

22

2B

V

erm

The mass spectrum of naturallyoccurring neon, showing threeisotopes.

Mass Spectrometer

21.4.1. For the mass spectrometer described in the text, the magnetic field is varied to allow ions of varying mass to reach the detector. A different method would be to maintain a constant magnetic field and vary the accelerating potential difference V. Assuming the magnetic field is held at B = 0.250 T and that r = 0.0750 m, in which of the following ranges of voltages could one detect both oxygen molecules (m = 2.656 1026 kg) and nitrogen molecules (m = 2.325 1026 kg).

a) 4200 to 4900 V

b) 3300 to 4500 V

c) 5100 to 6000 V

d) 2100 to 3100 V

e) 4800 to 5700 V

21.4.2. What must the initial state of motion of a charged particle be if it will follow a helical path in a magnetic field?

a) It must be moving at an angle that is neither parallel to nor perpendicular to the magnetic field.

b) It must be moving parallel to the magnetic field.

c) It must be moving perpendicular to the magnetic field.

d) It must be moving in the direction opposite to the magnetic field.

e) It must be initially at rest when it is placed in the magnetic field.



The Force on a Current in a Magnetic Field

Magnetic Force on Current-Carrying Wire F = I L B sin

I: current in Amps L: length in meters B: magnetic field in Tesla : angle between current and field

Derive this!

sinqvBF sinBtvt

qF

L

I

sinILBF sinqvBt

tF

The magnetic force on themoving charges pushes thewire to the right.

Force on Current Carrying Wire

Example 5 The Force and Acceleration in a LoudspeakerThe voice coil of a speaker has a diameter of 0.0025 m, contains 55 turns of wire, and is placed in a 0.10-T magnetic field. The current in the voice coil is 2.0 A. (a) Determine the magnetic force that acts on the coil and the cone. (b) The voice coil and cone have a combined mass of 0.0200 kg. Find their acceleration.

(a) sinILBF

(b)m

Fa

N 86.0F

90sinT 10.0m 0025.055A 0.2

2sm43kg 020.0

N 86.0

21.5.1. Three long, straight, identical wires are inserted one at a time into a magnetic field directed due east. Wire A carries a current of 2 A in the direction of 45 south of east. Wire B carries a current of 8 A, due north. Wire C carries a current of 10 A, due west. Rank the wires in terms of the magnitude of the magnetic force on each wire, with the largest force listed first and the smallest force listed last.

a) A > B > C

b) B > A > C

c) C > B > A

d) A > C > B

e) B > C > A

21.5.2. A portion of a loop of wire passes between the poles of a magnet as shown. We are viewing the circuit from above. When the switch is closed and a current passes through the circuit, what is the movement, if any, of the wire between the poles of the magnet?

a) The wire moves toward the north pole of the magnet.

b) The wire moves toward the south pole of the magnet.

c) The wire moves upward (toward us).

d) The wire moves downward (away from us).

e) The wire doesn’t move.



The Torque on a Current-Carrying Coil (Not AP-B)

The two forces on the loop have equal magnitude but an applicationof RHR-1 shows that they are opposite in direction.

Torque on Current Carrying Coil

The loop tends to rotate such that itsnormal becomes aligned with the magneticfield.


sinsinsinNet torque 21

21 IABwILBwILB

sin

momentmagnetic

BNIA

number of turns of wire


Example 6 The Torque Exerted on a Current-Carrying Coil

A coil of wire has an area of 2.0x10-4m2, consists of 100 loops or turns,and contains a current of 0.045 A. The coil is placed in a uniform magneticfield of magnitude 0.15 T. (a) Determine the magnetic moment of the coil.(b)Find the maximum torque that the magnetic field can exert on the coil.

(a)

2424

momentmagnetic

mA100.9m100.2A 045.0100 NIA

mN104.190sinT 15.0mA100.9sin 424

momentmagnetic

BNIA(b)

The basic components of a dc motor.


21.6.1. Small charged disks are inserted into a larger, insulating disk. A compass is placed near the larger disk and points due north as shown. The larger disk is then rotated uniformly counterclockwise (as viewed from above). What, if anything, will happen?

a) The north end of the compass will move toward the large diskas it rotates.

b) The north end of the compass will move away from the large disk as it rotates.

c) The compass will not be affected by the motion of the large disk.

d) The north end of the compass will oscillate toward and away from the large disk as it rotates.

21.6.2. A circular loop of wire is placed in a magnetic field such that the plane of the loop is perpendicular to the magnetic field. The loop is then connected to a battery and a current then flows through the loop. Which one of the following statements concerning this situation is true?

a) The magnetic force exerts a net torque on the loop.

b) The magnetic force exerts a net force on the loop.

c) The magnetic force exerts both a net force and a net torque on the loop.

d) The magnetic field has no affect on the loop.



Magnetic Fields Produced by Currents (Straight Wires Only)

Magnetic Fields…

Affect moving charge

F = qvBsin

F = ILBsin

Hand rule is used to determine direction of this force.

They are also caused by moving charge!

Right-Hand Rule No. 2. Curl the fingers of theright hand into the shape of a half-circle. Point the thumb in the direction of the conventional current, and the tips of the fingers will pointin the direction of the magnetic field.

Right Hand Rule for Straight Currents

A LONG, STRAIGHT WIRE

r

IB o

2

AmT104 7 o

permeability of free space

Magnitude of Magnetic Field

r : radius from center of wire

Example 7 A Current Exerts a Magnetic Force on a Moving ChargeThe long straight wire carries a current of 3.0 A. A particle has a charge of +6.5x10-6 C and is moving parallel to the wire at a distance of 0.050 m. The speed of the particle is 280 m/s.Determine the magnitude and direction of the magnetic force on the particle.

sinqvBF

r

IB o

2

sin2

r

IqvF o

oo

m

AsmCxF 0.90sin

050.02

0.3280105.6 6

NxF 8102.2

Current carrying wires can exert forces on each other.

Multiple Wires

Principle of Superposition

When there are two or more currents forming a magnetic

field, calculate B due to each current separately and then add

them together using vector addition.

Important Information: All other geometries noted in

book are “not” on the AP B test!

Conceptual Example 9 The Net Force That a Current-Carrying WireExerts on a Current Carrying CoilIs the coil attracted to, or repelled by the wire?

Coil would be attracted to the wire

A SINGLE LOOP OF WIRE

center of circular loop

R

IB o

2

Magnetic Field

Example 10 Finding the Net Magnetic FieldA long straight wire carries a current of 8.0 A and a circular loop of wire carries a current of 2.0 A and has a radius of 0.030 m. Find the magnitude and direction of the magnetic field at the center of the loop.

R

I

r

IB oo

2221

m 030.0

A 0.2

m 030.0

A 0.8

2

AmT104 7

B

2 2i

io

r

IB

R

I

r

Io 21

2

T101.1 5B

The field lines around the bar magnet resemble those around the loop.

Magnetic Fields Produced by Currents

Interior of a solenoid nIB o

number of turns perunit length

A Solenoid

A Cathode Ray Tube

21.7.1. The drawing represents a device called Roget’s Spiral. A coil of wire hangs vertically and its windings are parallel to one another. One end of the coil is connected by a wire to a terminal of a battery. The other end of the coil is slightly submerged below the surface of a cup of mercury. Mercury is a liquid metal at room temperature. The bottom of the cup is also metallic and connected by a wire to a switch. A wire from the switch to the battery completes the circuit. What is the behavior of this circuit after the switch is closed?

a) When current flows in the circuit, the coils of the wire move apart and the wire is extendedfurther into the mercury.

b) Nothing happens to the coil because there will not be a current in this circuit.

c) A current passes through the circuit until all of the mercury is boiled away.

d) When current flows in the circuit, the coils of the wire move together, causing the circuit to break at the surface of the mercury. The coil then extends and the process begins again when the circuit is once again complete.

21.7.2. Three very long, parallel wires (a small portion of each is shown in the drawing) are resting on a flat surface. The distance between wire B, which has a 15 mA current to the left, and its neighbors is 0.0015 m. Wire A carries a 10 mA current toward the right; and wire C carries a 5 mA current toward the right. Rank the wires in order of the magnitude of the net magnetic force on each, with the largest value first and the lowest value last.

a) A > B > C

b) B > A > C

c) C > B > A

d) A > C > B

e) B > C > A

21.7.3. The drawing shows a rectangular wire loop that has one side passing through the center of a solenoid. Which one of the following statements describes the force, if any, that acts on the rectangular loop when a current is passing through the solenoid.

a) The magnetic force causes the loop to move upward.

b) The magnetic force causes the loop to move downward.

c) The magnetic force causes the loop to move to the right.

d) The magnetic force causes the loop to move to the left.

e) The loop is not affected by the current passing through the solenoid or the magnetic field resulting from it.

21.7.4. The coils of a solenoid are stretched so that the length of the solenoid is twice its original length. Assuming the same current is passed though the solenoid before and after it is stretched, how does the magnetic field inside the solenoid change, if at all, as a result of the stretching?

a) The magnetic field after the stretching is one-fourth the value it was before stretching.

b) The magnetic field after the stretching is one-half the value it was before stretching.

c) The magnetic field after the stretching is the same as the value it was before stretching.

d) The magnetic field after the stretching is twice the value it was before stretching.

e) The magnetic field after the stretching is four times the value it was before stretching.

21.7.5. The equation for the magnetic field of a straight, current

carrying wire is given by , but the magnetic field at the center

of a single closed circular loop is given by . Although these

equations look similar, there is an important difference between these two equations, other that the factor of . What is it?

a) The µ0 factor is different for the two situations.

b) The variable R represents two different lengths.

c) The I represents two different types of current.

0

2

IB

r

0

2

IB

r

21.7.6. Complete the following statement: The magnetic field around a current-carrying, circular loop is most like that of

a) the Earth.

b) a current-carrying, rectangular loop

c) a short bar magnet.

d) a long, straight, current-carrying wire.

e) two long, straight wires that carry currents in opposite directions.

21.7.7. Two parallel wires have currents that have the same direction, but differing magnitude. The current in wire A is I; and the current in wire B is 2I. Which one of the following statements concerning this situation is true?

a) Wire A attracts wire B with half the force that wire B attracts wire A.

b) Wire A attracts wire B with twice the force that wire B attracts wire A.

c) Both wires attract each other with the same amount of force.

d) Wire A repels wire B with half the force that wire B attracts wire A.

e) Wire A repels wire B with twice the force that wire B attracts wire A.

21.7.8. Two parallel wires have currents that are in opposite directions and have differing magnitudes. The current in wire A is I; and the current in wire B is 2I. Which one of the following statements concerning this situation is true?

a) Wire A attracts wire B with half the force that wire B attracts wire A.

b) Wire A attracts wire B with twice the force that wire B attracts wire A.

c) Both wires attract each other with the same amount of force.

d) Wire A repels wire B with half the force that wire B attracts wire A.

e) Wire A repels wire B with twice the force that wire B attracts wire A.

21.7.9. The drawing shows two long, straight wires that are parallel to each other and carry a current of magnitude I toward you. The wires are separated by a distance d; and the centers of the wires are a distance d from the y axis. Which one of the following expressions correctly gives the magnitude of the total magnetic field at the origin of the x, y coordinate system?

a)

b)

c)

d)

e) zero tesla

0

2

i

d

0

2

i

d

0

2

i

d

0i

d



Ampère’s Law (Not AP-B)

AMPERE’S LAW FOR STATIC MAGNETIC FIELDS

For any current geometry that produces a magnetic field that does not change in time,

IB o ||

net currentpassing throughsurface bounded by path

Example 11 An Infinitely Long, Straight, Current-Carrying Wire

Use Ampere’s law to obtain the magnetic field.

IB o ||

IB o

IrB o 2

r

IB o

2



Magnetic Materials (Not AP-B)

The intrinsic “spin” and orbital motion of electrons gives rise to the magnetic properties of materials.

In ferromagnetic materials groups of neighboring atoms, forming magnetic domains, the spins of electrons are naturally aligned witheach other.

Magnetic Materials

Magnetic Materials

21.9.1. An initially unmagnetized iron bar is placed next to a solenoid. Which one of the following statements describes the iron bar after the solenoid is connected to the battery?

a) A magnetic force accelerates the bar to the right.

b) Since the bar is unmagnetized, there will not be any affect on the bar.

c) The magnetic field of the solenoid will cause a current to flow in a loop that extends from one end of the bar to the other and that continues until the battery is disconnected from the solenoid.

d) The magnetic field of the solenoid induces magnetism in the bar with the bar’s north pole nearest to the solenoid.

e) The magnetic field of the solenoid induces magnetism in the bar with the bar’s south pole nearest to the solenoid.

Ch 21 Magnetic Fields and Forces

Education

Transcript of Ch 21 Magnetic Fields and Forces